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Can someone explain the singular value decomposition of a tensor (maybe a 3 dimensional matrix) with an example? It is intuitively difficult to the get the meaning from just the formulas.

On a related note, consider the following scenario. There are n cars on the road and say the distance traveled by each of them is determined only by mood of the driver(let's think that higher the value the more he drives) and number of people in the car(which is independent to the mood of the driver).

If I were to model this as a 3 dimensional(distance traveled by the car, mood, number of people in car being the 3 dimensions) matrix and with some sufficient data points, can I get the distance traveled by a specific car given a mood and the number of people in the car(of course the data point is missing for this)?

The real use case I have probably has more dimensions(add hour of the day as a 4th dimension).

My problem is to exactly understand the process of decomposing the tensor to a singular matrix and also, given other (n-1) points, how do I get the distance traveled by the car?

Probably there are better approaches to solve this example, but I am specifically looking for this approach to understand higher order SVD.

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  • $\begingroup$ What is a "3D matrix"? A $3\times 3$ matrix? Or some kind of tensor? $\endgroup$ – user7530 Dec 19 '13 at 18:02
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    $\begingroup$ I meant 3 dimensional. I edited the question for more clarity. Thanks $\endgroup$ – user2489122 Dec 19 '13 at 18:13
  • $\begingroup$ Apparently, you'll have to be more specific about exactly what it is you're looking for. $\endgroup$ – Omnomnomnom Dec 19 '13 at 18:32
  • $\begingroup$ @Omnomnomnom: I have tried explaining it using a scenario. Hope it helps. Thanks $\endgroup$ – user2489122 Dec 19 '13 at 19:00
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Don't have enough points to add this as a comment...but you might want to look at "Tensor Train Decompositions" rather than "Higher Order SVDs".

http://spring.inm.ras.ru/osel/?p=121

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  • $\begingroup$ There are lot of theories such as SVD, Tucker, LU, QR. But most of them are in the form of publications which are for most parts incomprehensible for non-mathematicians like myself. I guess this field is still in research phase and yet to be used mainstream. $\endgroup$ – user2489122 Dec 20 '13 at 17:51

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